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Words like "therefore," "thus," and "however" are actually adverbs (a test for adverbs is if you can move them around in the clause: I am therefore full, I therefore am full, I am full, therefore). They don't act as conjunctions, so if you are joining two clauses, use a conjunction, or (because a semicolon can join two independent clauses) a semicolon.

It is true that such words are adverbs, but it is not true that you need a conjunction. Previously, a semicolon before, and a comma after would be required, but this is now considered obsolete. Simply preceding with a comma is sufficient.

It is acceptable to use a comma for short clauses (Chicago Manual of Style):

"I ate a burger, I'm full."

But that looks hideous on the page.

Commonly,

-Because... (therefore implied): Because I ate a hamburger, I'm full. (very clear)-That... (to be ("is") attribute): That I ate a hamburger, I'm quite full. (kinda prosaic)-If... then: If I ate a hamburger, then I'd be full. (more formal)-If... (then implied): If I ate a hamburger, I'd be full. (less formal but flows better in speech)-Since... : Since I ate a hamburger, I've been full. (time: I've been full since then.)-Since... (then implied): Since I ate a hamburger, I'm full. (logic: "since" used as an alternative to "because")-As... : As I ate a hamburger, I got full. (time and logic)-As... : As I'm hungry, I think I'll go eat a hamburger. (alternative to "because" and "since")

and many more, included inverted logic and time.

Thus is rarer (no pun intended) and tends to be used as a conclusion after a lengthy train of thought and is used to mean "in the manner indicated," most often found in instructions and is therefore less appropriate for logical expression:

Today, I ate so many things. I ate.... Therefore, I'm full.

This is how you could prepare a hamburger. First, put the beef on the grill... Thusly, you now have a feast.

All of these are transition words. They all show a result. The grammatical correct way to use them is as followed:I ate a hamburger, so I am full. I ate a hamburger; therefore, I am full.I ate a hamburger; thus, I am full.

Thus and therefore are interchangeable. Yet, "thus" is very formal. I would reserve it for academic papers.

@ps.nikki - Jennifer showed the "formal correct" punctuation (I'll go with Jasper on that - when you're writing informally, you can do what you like), but it's necessary to know the difference between (coordinating) conjunctions - 'so, but' etc and (conjunctive) adverbs - 'therefore, thus, however' etc - a good dictionary will tell you. So to build on what Jasper said:

Incidentally both 'therefore' and 'thus' are now quite often used after 'and' and without a semicolon.The first example is from Oxford Advanced Learner's Dictionary, the second from Oxford Dictionaries Online:

'He is the eldest son and thus heir to the title''He was injured and therefore unable to play'

Oxford Dictionaries Online also gives an example of 'thus' without semicolon or following comma, where it's followed by a participle rather than a full clause with subject and finite verb.

Burke knocked out Byrne, thus becoming champion

Personally I go by how long a pause sounds right, rather than by rigid rules. If you wouldn't pause in speech, don't use any punctuation. And use a comma where you'd normally pause briefly. Those last two sentences follow that principle, but also follow the formal "rule" - subordinate clause + comma + main clause, main clause + subordinate clause (without comma). And if you want a longer pause go for a semicolon.

Unlike in some languages, in English there is usually a direct connection between pronunciation and speech patterns. One thing I have to do with my Polish students when they're reading is to get them to pause at commas. When they do, the text usually becomes easier to understand.